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• W159618981 abstract "If ( bar{F} ) is an algebraic closure of a Galois field F, then for each integer n ≥ 1 there is exactly one subfield E n of ( bar{F} ) containing F and having degree n over F. For a prime number r, we consider the r-primary closure ( {bar{F}_{r}}: = bigcup {_{{m geqslant 0}}{E_{{{r^{m}}}}}} ) over F and prove, under the assumption that r ≥ 7, but without any restriction on the cardinality q of F, the existence of a universal generator for ( {bar{F}_{r}} ) over F: this is a sequence ( w = {({w_{{{r^{m}}}}})_{{m geqslant 0}}} ) in ( {bar{F}_{r}} ) which satisfies all the following properties: (1)( {w_{{{r^{m}}}}} ) is a. primitive element of ( {E_{{{r^{m}}}}} ) (for all m ≥ 0),(2)( {w_{{{r^{m}}}}} ) generates a normal basis for ( {E_{{{r^{m}}}}} ) over F (for all m ≥ 0),(3)w is norm-compatible,(4)w is trace compatible.We prove furthermore that (2) can be strengthened to (2 c ) ( {w_{{{r^{m}}}}} ) is completely free in ( {E_{{{r^{m}}}}} ) over F (for all m ≥ 0),which means that ( {w_{{{r^{m}}}}} ) simultaneously generates a normal basis for ( {E_{{{r^{m}}}}} ) over ( {E_{{{r^{i}}}}} ) for all i = 0, 1, …, m, whence w is called a complete universal generator for ( {bar{F}_{r}} ) over F. The results establish a (complete) primitive normal basis theorem for ( {bar{F}_{r}} ) over F." @default.
• W159618981 created "2016-06-24" @default.
• W159618981 creator A5002315634 @default.
• W159618981 date "2001-01-01" @default.
• W159618981 modified "2023-10-13" @default.
• W159618981 title "Universal Generators for Primary Closures of Galois Fields" @default.
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• W159618981 doi "https://doi.org/10.1007/978-3-642-56755-1_17" @default.
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